What is the limit as x approaches 0 of 8xcsc(9x) ?

The limit as x approaches 0 of 8xcsc(9x) is 8/9

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limit as x approaches 0 of 8xcsc(9x)

x \to 0 lim (8 x csc (9 x))

\frac{8}{9}

Decimal

0.88888 \dots

Solution steps

x \to 0 lim (8 x csc (9 x))

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 8 \cdot x \to 0 lim (x csc (9 x))

Express with sin, cos

= 8 \cdot x \to 0 lim (\frac{x}{sin ( 9 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( 9 x ) \cdot 9})

= 8 \cdot x \to 0 lim (\frac{1}{cos ( 9 x ) \cdot 9})

Plug in the value

x = 0

= 8 \cdot \frac{1}{cos ( 9 \cdot 0 ) \cdot 9}

Simplify

8 \cdot \frac{1}{cos ( 9 \cdot 0 ) \cdot 9} : \frac{8}{9}

= \frac{8}{9}

x \to 2 lim (\frac{x - 1}{x ^{3} - 1})

x \to 0 lim (\frac{\frac{2}{3} cos ( x ) - 1}{x})

x \to 0 lim (\frac{e ^{x} - 1}{5 x ^{2}})

x \to 0 lim (\frac{\frac{1}{x ^{2}}}{e ^{\frac{1}{x ^{2}}}})

x \to 0 lim (x^{8})

What is the limit as x approaches 0 of 8xcsc(9x) ?
The limit as x approaches 0 of 8xcsc(9x) is 8/9